How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced — the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted — a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated — a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
A group homomorphism is injective if and only if its kernel is trivial
Statement
A group homomorphism is injective if and only if its kernel is trivial.
For a group homomorphism , is injective exactly when .
Facts & Assumptions
Given: A group homomorphism .
Homomorphisms preserve inverses and products (A group homomorphism automatically satisfies and , and for every ; for monoid homomorphisms preservation of the identity must be assumed).
Injectivity means that equal values have equal arguments (Injection, surjection, bijection).
In a group, implies (In a group , and , the order of the last product being essential).
Proof
If is injective and , then , so .
Conversely, if and , then , whence and .
The two implications prove the equivalence.
Depends on
- The kernel and image of a group homomorphism
- A group homomorphism automatically satisfies $f(e) = e'$ and $f(g^{-1}) = f(g)^{-1}$, and $f(g^{n}) = f(g)^{n}$ for every $n \in \mathbb{Z}$; for monoid homomorphisms preservation of the identity must be assumed
- Injection, surjection, bijection
- In a group $e^{-1} = e$, $(g^{-1})^{-1} = g$ and $(gh)^{-1} = h^{-1}g^{-1}$, the order of the last product being essential
Used by
- Every canonical factor map into a free product is injective Corollary
- The factor maps into a free product with amalgamation are injective Corollary
- The doubling endomorphism of (ℤ,+) has trivial kernel but is not surjective Counterexample
- Free products with amalgamation along monomorphisms Definition
- Cayley's theorem: every group G is isomorphic to a subgroup of Sym(G) Theorem
- First isomorphism theorem for groups: G/ker fcongimf Theorem
- Internal direct products are external direct products, equivalently every element has a unique factorisation Theorem
- Kernels and images of module homomorphisms are submodules, and injectivity is equivalent to trivial kernel Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 51 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Ernst, An Inquiry-Based Approach to Abstract Algebra, Homomorphisms (standard reference, not scraped)